There are different rules for use of images, depending on the licensing of the image, and how you intend to use it.
Public Domain or CC0 or CC PDM
Under US copyright law, all works published before 1927 are in the public domain, meaning you can use and even modify the work. So pay attention to dates on the images.
The owner of the image can also choose to tag the image as being in the public domain and let anyone use it. The most common way of doing this online is to add a Creative Commons public domain license, either CC0 or CC PDM (public domain mark). See https://creativecommons.org/publicdomain/ for more information.
Other Creative Commons Licenses
There are 6 other licenses, allowing different levels of remixing, depending on whether it's non-commercial, if you also allow for remixing, etc.
After public domain or CC0, the most open one is CC By, allowing you to reuse or remix, as long as you give attribution, or give credit to the original creation. Which you need to do anyway to cite the image. See https://creativecommons.org/licenses/by/4.0/ for more information.
What about images that aren't public domain or Creative Commons licensed?
Those images are copyrighted, and the owner of the image can charge for use, or even sue you for using the image without permission.
If they put the image online, don't they expect people to download and use them?
Those companies that are selling images do put watermarks on the images or technologically restrict downloading. But the law doesn't put the burden on the copyright owners to do this. The burden is on you, the user of the image, to use is legally.
But I'm using it for educational purposes, not to make money.
One major exception to copyright protections for works is "fair use," and one of the four factors as to whether it is fair use is using it for "nonprofit educational purposes." Unfortunately, it isn't as helpful as you would hope, because it isn't a blanket exception to copyright. Instead, it is quite complicated and vague as to whether a particular use falls under fair use.
But I'm citing the image.
Citing the image is to prevent you from plagiarizing, not a blanket protection from copyright.
If you can find public domain or Creative Commons licensed images, you ought use those, rather than relying on fair use.
So what is fair use?
The law states:
17 U.S. Code § 107 - Limitations on exclusive rights: Fair use
Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include—
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.
Examples of the four factors
(1) Educational purposes is a factor, but not the sole factor. But it is in favor of it being fair use. Acknowledging that the image is copyrighted and warning others of that if they want to subsequently use the image is also looked on favorably. But just citing the image does not equate to fair use.
(2) The courts usually support stricter protections for creative works more than factual, nonfiction ones.
(3) Using a thumbnail of a copyrighted work is seen as more favorable to fair use than downloading the original and not modifying it.
(4) If the copyright owner is clearly selling the image, and you choose not to pay, then it is much less likely to fall under fair use.
For other detailed examples, see the Visual Resources Association's Digital Image Rights Computator, and their Statement on the Fair Use of Images for Teaching, Research, and Study (2013).